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In this book, three authors introduce readers to strong approximation methods, analytic pro-p groups and zeta functions of groups. Each chapter illustrates connections between infinite group theory, number theory and Lie theory. The first introduces the theory of compact p-adic Lie groups. The second explains how methods from linear algebraic groups can be utilised to study the finite images of linear groups. The final chapter provides an overview of zeta functions associated to groups and rings. Derived from an LMS/EPSRC Short Course for graduate students, this book provides a concise introduction to a very active research area and assumes less prior knowledge than existing monographs or original research articles. Accessible to beginning graduate students in group theory, it will also appeal to researchers interested in infinite group theory and its interface with Lie theory and number theory.

Profinite groups. --- Group theory. --- Group theory --- Profinite groups --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra

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The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. The book is reasonably self-contained. Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. In addition to basic facts about general profinite groups, the book emphasizes free constructions (particularly free profinite groups and the structure of their subgroups). Homology and cohomology is described with a minimum of prerequisites. This second edition contains three new appendices dealing with a new characterization of free profinite groups, presentations of pro-p groups and a new conceptually simpler approach to the proof of some classical subgroup theorems. Throughout the text there are additions in the form of new results, improved proofs, typographical corrections, and an enlarged bibliography. The list of open questions has been updated; comments and references have been added about those previously open problems that have been solved after the first edition appeared.

Profinite groups. --- Profinite groups --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Group theory. --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematics. --- Topological groups. --- Lie groups. --- Number theory. --- Topology. --- Group Theory and Generalizations. --- Topological Groups, Lie Groups. --- Number Theory. --- Group theory --- Topological Groups. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Number study --- Numbers, Theory of --- Groups, Topological --- Continuous groups --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups